The function may be called by the names: g07dcc, nag_univar_robust_1var_mestim_wgt or nag_robust_m_estim_1var_usr.
3Description
The data consists of a sample of size $n$, denoted by ${x}_{1},{x}_{2},\dots ,{x}_{n}$, drawn from a random variable $X$.
The ${x}_{i}$ are assumed to be independent with an unknown distribution function of the form,
$$F(({x}_{i}-\theta )/\sigma )$$
where $\theta $ is a location parameter, and $\sigma $ is a scale parameter. $M$-estimators of $\theta $ and $\sigma $ are given by the solution to the following system of equations;
where $\psi $ and $\chi $ are user-supplied weight functions, and $\beta $ is a constant. Optionally the second equation can be omitted and the first equation is solved for $\hat{\theta}$ using an assigned value of $\sigma ={\sigma}_{c}$.
The constant $\beta $ should be chosen so that $\hat{\sigma}$ is an unbiased estimator when ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$ has a Normal distribution. To achieve this the value of $\beta $ is calculated as:
The initial values for $\hat{\theta}$ and $\hat{\sigma}$ may be user-supplied or calculated within g07dbc as the sample median and an estimate of $\sigma $ based on the median absolute deviation respectively.
g07dcc is based upon function LYHALG within the ROBETH library, see Marazzi (1987).
4References
Hampel F R, Ronchetti E M, Rousseeuw P J and Stahel W A (1986) Robust Statistics. The Approach Based on Influence Functions Wiley
Huber P J (1981) Robust Statistics Wiley
Marazzi A (1987) Subroutines for robust estimation of location and scale in ROBETH Cah. Rech. Doc. IUMSP, No. 3 ROB 1 Institut Universitaire de Médecine Sociale et Préventive, Lausanne
5Arguments
1: $\mathbf{chi}$ – function, supplied by the userExternal Function
chi must return the value of the weight function $\chi $ for a given value of its argument. The value of $\chi $ must be non-negative.
On entry: the argument for which chi must be evaluated.
2: $\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to chi.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling g07dcc you may allocate memory and initialize these pointers with various quantities for use by chi when called from g07dcc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note:chi should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by g07dcc. If your code inadvertently does return any NaNs or infinities, g07dcc is likely to produce unexpected results.
2: $\mathbf{psi}$ – function, supplied by the userExternal Function
psi must return the value of the weight function $\psi $ for a given value of its argument.
On entry: the argument for which psi must be evaluated.
2: $\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to psi.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling g07dcc you may allocate memory and initialize these pointers with various quantities for use by psi when called from g07dcc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note:psi should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by g07dcc. If your code inadvertently does return any NaNs or infinities, g07dcc is likely to produce unexpected results.
3: $\mathbf{isigma}$ – IntegerInput
On entry: the value assigned to isigma determines whether $\hat{\sigma}$ is to be simultaneously estimated.
${\mathbf{isigma}}=0$
The estimation of $\hat{\sigma}$ is bypassed and sigma is set equal to ${\sigma}_{c}$.
On entry: the vector of observations, ${x}_{1},{x}_{2},\dots ,{x}_{n}$.
6: $\mathbf{beta}$ – doubleInput
On entry: the value of the constant $\beta $ of the chosen chi function.
Constraint:
${\mathbf{beta}}>0.0$.
7: $\mathbf{theta}$ – double *Input/Output
On entry: if ${\mathbf{sigma}}>0$, theta must be set to the required starting value of the estimate of the location parameter $\hat{\theta}$. A reasonable initial value for $\hat{\theta}$ will often be the sample mean or median.
On exit: the $M$-estimate of the location parameter $\hat{\theta}$.
8: $\mathbf{sigma}$ – double *Input/Output
On entry: the role of sigma depends on the value assigned to isigma as follows.
If ${\mathbf{isigma}}=1$, sigma must be assigned a value which determines the values of the starting points for the calculation of $\hat{\theta}$ and $\hat{\sigma}$. If ${\mathbf{sigma}}\le 0.0$, g07dcc will determine the starting points of $\hat{\theta}$ and $\hat{\sigma}$. Otherwise, the value assigned to sigma will be taken as the starting point for $\hat{\sigma}$, and theta must be assigned a relevant value before entry, see above.
If ${\mathbf{isigma}}=0$, sigma must be assigned a value which determines the values of ${\sigma}_{c}$, which is held fixed during the iterations, and the starting value for the calculation of $\hat{\theta}$. If ${\mathbf{sigma}}\le 0$, g07dcc will determine the value of ${\sigma}_{c}$ as the median absolute deviation adjusted to reduce bias (see g07dac) and the starting point for $\theta $. Otherwise, the value assigned to sigma will be taken as the value of ${\sigma}_{c}$ and theta must be assigned a relevant value before entry, see above.
On exit: the $M$-estimate of the scale parameter $\hat{\sigma}$, if isigma was assigned the value $1$ on entry, otherwise sigma will contain the initial fixed value ${\sigma}_{c}$.
9: $\mathbf{maxit}$ – IntegerInput
On entry: the maximum number of iterations that should be used during the estimation.
Suggested value:
${\mathbf{maxit}}=50$.
Constraint:
${\mathbf{maxit}}>0$.
10: $\mathbf{tol}$ – doubleInput
On entry: the relative precision for the final estimates. Convergence is assumed when the increments for theta, and sigma are less than ${\mathbf{tol}}\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1.0,{\sigma}_{k-1})$.
On exit: the number of iterations that were used during the estimation.
13: $\mathbf{comm}$ – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
14: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_FUN_RET_VAL
The chi function returned a negative value: ${\mathbf{chi}}=\u27e8\mathit{\text{value}}\u27e9$.
NE_INT
On entry, ${\mathbf{isigma}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{isigma}}=0$ or $1$.
On entry, ${\mathbf{maxit}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{maxit}}>0$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}>1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{beta}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{beta}}>0.0$.
On entry, ${\mathbf{tol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{tol}}>0.0$.
Current estimate of sigma is zero or negative: ${\mathbf{sigma}}=\u27e8\mathit{\text{value}}\u27e9$. This error exit is very unlikely, although it may be caused by too large an initial value of sigma.
NE_TOO_MANY_ITER
Number of iterations required exceeds maxit: ${\mathbf{maxit}}=\u27e8\mathit{\text{value}}\u27e9$.
NE_ZERO_RESID
All winsorized residuals are zero. This may occur when using the ${\mathbf{isigma}}=0$ option with a redescending $\psi $ function, i.e., Hampel's piecewise linear function, Andrew's sine wave, and Tukey's biweight.
If the given value of $\sigma $ is too small, the standardized residuals $\frac{{x}_{i}-{\hat{\theta}}_{k}}{{\sigma}_{c}}$, will be large and all the residuals may fall into the region for which $\psi \left(t\right)=0$. This may incorrectly terminate the iterations thus making theta and sigma invalid.
Re-enter the function with a larger value of ${\sigma}_{c}$ or with ${\mathbf{isigma}}=1$.
7Accuracy
On successful exit the accuracy of the results is related to the value of tol, see Section 5.
8Parallelism and Performance
g07dcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g07dcc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
When you supply the initial values, care has to be taken over the choice of the initial value of $\sigma $. If too small a value is chosen then initial values of the standardized residuals $\frac{{x}_{i}-{\hat{\theta}}_{k}}{\sigma}$ will be large. If the redescending $\psi $ functions are used, i.e., $\psi =0$ if $\left|t\right|>\tau $, for some positive constant $\tau $, then these large values are Winsorized as zero. If a sufficient number of the residuals fall into this category then a false solution may be returned, see page 152 of Hampel et al. (1986).
10Example
The following program reads in a set of data consisting of eleven observations of a variable $X$.
The psi and chi functions used are Hampel's Piecewise Linear Function and Hubers chi function respectively.
Using the following starting values various estimates of $\theta $ and $\sigma $ are calculated and printed along with the number of iterations used:
(a)g07dcc determined the starting values, $\sigma $ is estimated simultaneously.
(b)You must supply the starting values, $\sigma $ is estimated simultaneously.
(c)g07dcc determined the starting values, $\sigma $ is fixed.
(d)You must supply the starting values, $\sigma $ is fixed.